GRADINGS OF SIMPLE JORDAN ALGEBRAS AND THEIR RELATION TO THE GRADINGS OF SIMPLE ASSOCIATIVE ALGEBRAS

In this paper we describe all group gradings of the simple Jordan algebra of a non-degenerate symmetric form on a vector space over a field of characteristic different from 2. If we use the notion of the Clifford algebra, then we are able to recover some of the gradings on matrix algebras obtained in an entirely different way in [BSZ] Bahturin, Y., Seghal, S. and Zaicev, M. in press. Group Gradings of Associative Algebras. J. Algebra, [Web of Science ®] , [Google Scholar].

     

Asymptotic results for finite superpositions of Ornstein–Uhlenbeck processes

A model of intermittency based on superposition of Lévy driven Ornstein–Uhlenbeck processes is studied in [6 Grahovac, D., Leonenko, N., Sikorskii, A., and Te?niak, I. 2016. Intermittency of superpositions of Ornstein–Uhlenbeck type processes. J. Stat. Phys. 165:390–408.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, as shown in Theorem 5.1 in that paper, finite superpositions obey a (sample path) central limit theorem under suitable hypotheses. In this paper we prove large (and moderate) deviation results associated with this central limit theorem.     

GORENSTEIN TILED ORDERS

Let Λ = {O, E(Λ)} be a reduced tiled Gorenstein order with Jacobson radical R and J a two-sided ideal of Λ such that Λ ? R ² ? J ? Rⁿ (n ≥ 2). The quotient ring Λ/J is quasi-Frobenius (QF) if and only if there exists p ∈ R ² such that J = pΛ = Λp. We prove that an adjacency matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a finite group G is an exponent matrix of a reduced Gorenstein tiled order if and only if G = G_k = (2) × ? × (2).

Commutative Gorenstein rings appeared at first in the paper [3] Gorenstein, D. 1952. An Arithmetic Theory of Adjoint Plane Curves. Trans. AMS., 72: 414–436. [Crossref], [Web of Science ®] , [Google Scholar]. Torsion-free modules over commutative Gorenstein domains were investigated in [1] Bass, H. 1963. On the Ubiquity of Gorenstein Rings. Math. Z., 82(1): 8–27. [Crossref] , [Google Scholar]. Noncommutative Gorenstein orders were considered in [2] Drozd, Yu. A., Kirichenko, V. V. and Roiter, A. V. 1967. On Hereditary and Bass Orders. Izv. Akad. Nauk SSSR Ser. Mat., 31: 1415–1436. Math. USSR – Izvestija, 1967, 1, 1357–1375 [Google Scholar] and [10] Roggenkamp, K. W. 1970. Lattices Over Orders II Berlin, Heidelberg, New York: Springer-Verlag. [Crossref] , [Google Scholar]. Relations between Gorenstein orders and quasi-Frobenius rings were studied in [5] Kirichenko, V. V. 1978. On Quasi-Frobenius Rings and Gorenstein Orders. Trudy Math. Steklov Inst., 148: 168–174. (in Russian) [Google Scholar]. Arbitrary tiled orders were considered in [4] Jategaonkar, V. A. 1974. Global Dimension of Tiled Orders Over a Discrete Valuation Ring. Trans. AMS., 196: 313–330. [Crossref], [Web of Science ®] , [Google Scholar], 11-14 Simson, D. 1992. Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra, Logic and Appl. Vol. 4, Gordon and Breach Science Publishers. Zavadskij, A. G. 1973. The Structure of Orders with Completely Decomposable Representations. Mat. Zametki, 13: 325–335. (in Russian) Zavadskij, A. G. and Kirichenko, V. V. 1976. Torsion-free Modules over Prime Rinqs. Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 57: 100–116. J. Soviet. Math. 1979, 11, 598–612 Zavadskij, A. G. and Kirichenko, V. V. 1977. Semimaximal Rings of Finite Type. Mat. Sbornik, 103(No. 3): 323–345. Math. USSR Sbornik, 1977, 32 (3), 273–291 .     

UNIFORM FILTERS AND SEMI-COMPATIBILITY

In [5] García Román, M., Márquez Hernández, M. and Verschoren, A. 1997. Structure Sheaves and Noncommutative Topologies. J. of Algebra, 194: 224–244. [Crossref], [Web of Science ®] , [Google Scholar] we used functors which are compositions of localization functors to construct sheaves over an arbitrary ring R. These functors share some properties with localization, and questions like when is the composition of localizations a localization functor? arise naturally. In this note we answer this question and some related ones using the key concept of semi-compatibility.

     

ON FREE PRODUCTS OF ABSOLUTE GALOIS GROUPS. II

In [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar], the author gave a positive solution to the problem in the survey of Jarden [2] Jarden, M. 1996. “Infinite Galois Theory”. In Handbook of Algebra I Amsterdam: Elsevier Sci.. [Crossref] , [Google Scholar] on the closedness of the class of profinite groups that are isomorphic to absolute Galois groups of fields with respect to finite free products. In [3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar], O. V. Mel'nikov solved this problem for separable profinite groups ([3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar] was done earlier than [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar]). In the same case, a more exact result on the absolute Galois groups of fields of fixed characteristic was obtained there. The proof proposed in 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.   is simpler than that in [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar] and, in addition, provides the results of Mel'nikov.

On February, 2000, the author (knowing nothing about 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.  ) found one more proof of these results. In the author opinion, this proof is the simplest and the construction used in the proof, as well as its properties (cf. Propositio n 1) can have other applications.     

ON THE NORM OF THE FOURIER-JACOBI PROJECTION

We extend the results of Pollard [7] Pollard, H. 1949. The mean convergence of orthogonal series. III. Duke Math. J., 16: 189–191. [Crossref], [Web of Science ®] , [Google Scholar] and give asymptotic estimates for the norm of the Fourier-Jacobi projection operator in the appropriate weighted L_p space.     

ON THE STRONG LAW OF LARGE NUMBERS FOR SUMS OF INDEPENDENT BANACH SPACE VALUED RANDOM ELEMENTS

For a sequence of independent random elements {V _n ,n≥1} in a real separable Banach space X, necessary and, separately, sufficient conditions are provided for the strong law of large numbers ∑ _i=1 ⁿ (V _i ?c _i )/b _n →0 almost certainly to hold where {c _n ,n≥1} and {b _n >0,n≥1} are suitable sequences of centering elements in X and norming constants, respectively. The necessity result extends a real line result of Martikainen[14] Martikainen, A.I. 1979. On Necessary and Sufficient Conditions for the Strong Law of Large Numbers. Teor. Veroyatnost. i Primen., 24: 814–820. (English translation in Theory Probabl. Appl., 24 (1979) 820–823) [Google Scholar] to a Banach space setting. The sufficiency result assumes that X is of Rademacher type p (1≤p≤2) and is new even when X is the real line. It is general enough to include as special cases a strong law of Adler, Rosalsky, and Taylor[2] Adler, A., Rosalsky, A. and Taylor, R.L. 1989. Strong Laws of Large Numbers for Weighted Sums of Random Elements in Normed Linear Spaces. Int. J. Math. Math. Sci., 12: 507–530. [Crossref] , [Google Scholar] for sums of independent and identically distributed random elements and a strong law of Heyde[9] Heyde, C.C. 1968. On almost sure convergence for sums of independent random variables. Sankhya¯ Ser. A, 30: 353–358.  [Google Scholar] for sums of independent (real-valued) random variables. Illustrative examples are provided showing that the results are sharp and an example is presented satisfying the hypotheses of the sufficiency result but not those of Heyde's[9] Heyde, C.C. 1968. On almost sure convergence for sums of independent random variables. Sankhya¯ Ser. A, 30: 353–358.  [Google Scholar] theorem.     

Large time behavior of solutions of a dissipative semilnear heat equation

We study the current of the Pauli operator in a strong constant magnetic field. We prove that in the semi-classical limit the persistent current and the current from the interaction of the spin with the magnetic field cancel, in the case where the magnetic field is very strong. Furthermore we calculate the next term in the asymptotics and estimate the error. Finally, we discuss the connection between this work and the semi-classical estimate of the energy in strong magnetic fields proved by Lieb, Solovej and Yngvason [8] Lieb, E., Solovej, J. P. and Yngvason, J. 1994. Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions. Commun. Math. Phys., 161: 77–124. [Crossref], [Web of Science ®] , [Google Scholar].

     

THE NICHOLS ZOELLER THEOREM FOR HOPF ALGEBRAS IN THE CATEGORY OF YETTER DRINFELD MODULES

In 1989 Nichols and Zoeller [NZ] Nichols, W. D. and Zoeller, M. B. 1989. A Hopf algebra freeness theorem. Amer. J. Math., 111: 381–385. [Crossref], [Web of Science ®] , [Google Scholar] showed that finite dimensional k-Hopf algebras are free over Hopf subalgebras. An analog result for Yetter Drinfeld Hopf algebras was not known. In this paper the existence of such a basis will be proved. Moreover the existence of a basis in a certain categorial sense cannot be expected.     

GRADE AND GORENSTEIN DIMENSION

Let R be a commutative Noetherian ring and let M be a finite (that is, finitely generated) R-module. The notion grade of M, grade M, has been introduced by Rees as the least integer t ≥ 0 such that Ext ^t _R (M,R) ≠ 0, see [11] Rees, D. 1957. The Grade of an Ideal or Module. Proc. Camb. Phil. Soc., 53: 28–42. [Crossref] , [Google Scholar]. The Gorenstein dimension of M, G-dim M, has been introduced by Auslander as the largest integer t ≥ 0 such that Ext ^t _R (M, R) ≠ 0, see [3] Auslander, M. 1967. Anneaux De Gorenstein Et Torsion En Algebre Commutative Edited by: Mangeney, M., Peskine, C. and Szpiro, L. Paris: Ecole Normale Superieure de Jeunes Filles.  [Google Scholar]. In this paper the R-module M is called G-perfect if grade M = G-dim M. It is a generalization of perfect module. We prove several results for the new concept similar to the classical results.     

Grouped Functional Time Series Forecasting: An Application to Age-Specific Mortality Rates

Age-specific mortality rates are often disaggregated by different attributes, such as sex, state, and ethnicity. Forecasting age-specific mortality rates at the national and sub-national levels plays an important role in developing social policy. However, independent forecasts at the sub-national levels may not add up to the forecasts at the national level. To address this issue, we consider reconciling forecasts of age-specific mortality rates, extending the methods of Hyndman et al. in 2011 Hyndman, R. J., Ahmed, R. A., Athanasopoulos, G., and Shang, H. L. (2011), “Optimal Combination Forecasts for Hierarchical Time Series,” Computational Statistics and Data Analysis, 55, 2579–2589.[Crossref], [Web of Science ®] , [Google Scholar] to functional time series, where age is considered as a continuum. The grouped functional time series methods are used to produce point forecasts of mortality rates that are aggregated appropriately across different disaggregation factors. For evaluating forecast uncertainty, we propose a bootstrap method for reconciling interval forecasts. Using the regional age-specific mortality rates in Japan, obtained from the Japanese Mortality Database, we investigate the one- to ten-step-ahead point and interval forecast accuracies between the independent and grouped functional time series forecasting methods. The proposed methods are shown to be useful for reconciling forecasts of age-specific mortality rates at the national and sub-national levels. They also enjoy improved forecast accuracy averaged over different disaggregation factors. Supplementary materials for the article are available online.     

Calderón problem for connections

In this paper, we consider the problem of identifying a connection ? on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian ?*? over conformally transversally anisotropic (CTA) manifolds. This was proved in [9 Dos Santos Ferreira, D., Kenig, C., Salo, M., Uhlmann, G. (2009). Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178:119–171.[Crossref], [Web of Science ®] , [Google Scholar]] for line bundles in the case of the transversal manifold being simple—we generalize this result to the case where the transversal manifold only has an injective ray transform. Moreover, the construction of suitable Gaussian beam solutions on vector bundles is given for the case of the connection Laplacian and a potential, following the works of [11 Dos Santos Ferreira, D., Kurylev, Y., Lassas, M., Salo, M. (2016). The Calderón problem in transversally anisotropic geometries. J. Eur. Math. Soc., 18:2579–2626.[Crossref], [Web of Science ®] , [Google Scholar]]. This in turn enables us to construct the Complex Geometrical Optics (CGO) solutions and prove our main uniqueness result. We also reduce the problem to a new non-abelian X-ray transform for the case of simple transversal manifolds and higher rank vector bundles. Finally, we prove the recovery of a flat connection in general from the DN map, up to gauge equivalence, using an argument relating the Cauchy data of the connection Laplacian and the holonomy.     

Non–existence of theta–shaped self–similarly shrinking networks moving by curvature

We prove that there are no networks homeomorphic to the Greek “Theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees, such that under the motion by curvature they are self–similarly shrinking.

This fact completes the classification of the self–similarly shrinking networks in the plane with at most two triple junctions, see [5 Chen, X., Guo, J.-S. (2007). Self-similar solutions of a 2-D multiple-phase curvature flow. Phys. D. 229(1):22–34.[Crossref], [Web of Science ®] , [Google Scholar], 10 Hättenschweiler, J. (2007). Mean curvature flow of networks with triple junctions in the plane. Master’s thesis. ETH Zürich. [Google Scholar], 25 Schnürer, O. C., Azouani, A., Georgi, M., Hell, J., Nihar, J., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F., Smith, B. (2011). Evolution of convex lens–shaped networks under the curve shortening flow. Trans. Am. Math. Soc. 363(5):2265–2294.[Crossref], [Web of Science ®] , [Google Scholar], 2 Baldi, P., Haus, E., Mantegazza, C. (2016). Networks self-similarly moving by curvature with two triple junctions. Networks self-similarly moving by curvature with two triple junctions. 28(2017):323–338. [Google Scholar]].     

A NONCLASSICAL LAW OF THE ITERATED LOGARITHM FOR I.I.D. SQUARE INTEGRABLE RANDOM VARIABLES. II

Let {X _n , n≥1} be a sequence of i.i.d. random variable with EX ₁=0 and E X ₁ ²=1 and let {b _n , n≥1} be a sequence of positive constants monotonically approaching infinity such that lim inf _n→∞ b _n /log log n=1. It is proved that lim sup _n→∞ ∑ _i=1 ⁿ X _i /√2nb _n =1 almost certainly thereby extending the work of Klesov and Rosalsky[4] Klesov, O. and Rosalsky, A. 2001. A Nonclassical Law of the Iterated Logarithm for I.I.D. Square Integrable Random Variables. Stoch. Anal. Appl., 19: 627–641. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] to a much larger class of sequences {b _n , n≥1}. The tools employed in the argument are results of Bulinskii[1] Bulinskii, A.V. 1977. On Normalization in the Law of the Iterated Logarithm. Teor. Veroyatnost. i Primen., 22: 407–409. In Russian, English translation in Theory Probab. Appl., 22 (1977), 398–399 [Google Scholar] and Feller[2] Feller, W. 1943. The General Form of the So-Called Law of the Iterated Logarithm. Trans. Am. Math. Soc., 54: 373–402.  [Google Scholar] and the Strassen[5] Strassen, V. 1964. An Invariance Principle for the Law of the Iterated Logarithm. Z. Wahrsch. Verw. Gebiete, 3: 211–226. [Crossref] , [Google Scholar] strong invariance principle.     

Evaluating a model of forest succession using fuzzy analysis

In this paper we discuss how to select the optimal policy from a set of possible policies for a model of forest succession, which can be characterized by a set of trees and the corresponding average life-span with each possible tree transition. The transition probabilities are estimated by counting the numbers of sapling trees of each species under a canopy tree. [1] Horn, Henry S. 1975. Forest Succession. Sci. Amer., : 90–98.  [Google Scholar]. In our setting the transition matrix is defined by using the linguistic terms and as a consequence, the expected longevity of each tree is fuzzy. We use the Dempster–Shafer theory [8] Shafer, G. 1976. A Mathematical Theory of Evidence Princeton University Press.  [Google Scholar] ('76) together with techniques of Norton [7] Norton, J. 1988. Limit Theorems for Dempster's Rule of Combination. Theory and Decision, 25(3): 287–313. [Crossref], [Web of Science ®] , [Google Scholar] ('88) and Smetz [9] Smetz, P. 1990. Belief Functions versus Probability Functions. Uncertainty in Artificial Intelligence, 5: 1–8.  [Google Scholar] ('76) to approximate the transition probabilities.     

The infinitesimal and the little q-Schur superalgebras

In this paper, based on the results in [8 Du, J., Gu, H.-X. (2014). A realization of the quantum supergroup U(𝔤𝔩_m|n). J. Algebra 404:60–99.[Web of Science ®] , [Google Scholar]] we give a monomial basis for q-Schur superalgebra and then a presentation for it. The presentation is different from that in [12 El Turkey, H., Kujawa, J. (2012). Presenting Schur superalgebras. Pacific J. Math., 262(2):285–316.[Crossref], [Web of Science ®] , [Google Scholar]]. Imitating [3 Cox, A. G. (1997). On some applications of infinitesimal methods to quantum groups and related algebras. Ph.D. Thesis. University of London. [Google Scholar]] and [7 Du, J., Fu, Q., Wang, J.-P. (2005). Infinitesimal quantum 𝔤𝔩_n and little q-Schur algebras. J. Algebra 287:199–233.[Crossref], [Web of Science ®] , [Google Scholar]], we define the infinitesimal and the little q-Schur superalgebras. We give a “weight idempotent presentation” for infinitesimal q-Schur superalgebras. The BLM bases and monomial bases of little q-Schur superalgebras are obtained, and dimension formulas of infinitesimal and little q-Schur superalgebras are deduced.     

Splitting of operations for alternative and Malcev structures

In this paper, we define pre-Malcev algebras and alternative quadri-algebras and prove that they generalize pre-Lie algebras and quadri-algebras, respectively, to the alternative setting. We use the results and techniques from [4 Bai, C., Bellier, O., Guo, L., Ni, X. (2013). Splitting of operations, Manin products, and Rota-Baxter operators. Int. Math. Res. Not. 2013(3):485–524. [Google Scholar], 14 Gubarev, V. Y., Kolesnikov, P. S. (2013). Embedding of dendriform algebras into Rota-Baxter algebras. Cent. Eur. J. Math. 11(2):226–245.[Crossref], [Web of Science ®] , [Google Scholar]] to discuss and give explicit computations of different constructions in terms of bimodules, splitting of operations, and Rota–Baxter operators.